I think VOY - Counterpoint might fit the bill.

Reasons:

1. It's got a double double-cross plot with pretty high stakes.
2. It's traditional in the "alien of the week" format.
3. Janeway and Inspector Kashyk have excellent chemistry
4. Hefty abuse of the pattern buffer
5. Basically a self-contained episode that's largely confined to the ship

My first thought whenever the topic of what episode a person should to introduce Trek to someone comes up is "The Measure of a Man". Though perhaps a courtroom drama, while certainly something Trek dabbles with on multiple occasions, is not typical enough to fall under the umbrella of conventional.

Maybe something like "Children of the Comet" from season one of SNW. There's a strange mystery that's going to spell disaster for a pre-warp civilization, an alien of the week antagonist whom the Enterprise crew needs to figure out how to deal with without getting into a fight, and everything's neatly wrapped up by the end. The biggest mark against it would be the subplot where Pike's dealing with the knowledge that he's going to end up in a beep chair.

I wanna say “the corbomite maneuver” from TOS. I think it’s one of the best episodes of all, but most people don’t so i think it’s okay from the its all downhill perspective. It has everything: Kirk mentoring, an extremely powerful alien of the week, solving problems without fighting, but still tense situations. Such a great and representative episode.

From TNG maybe First Contact. Good intro episode as you get intro to the federation through the eyes of an alien of the week. The alien-chasing woman may be not a good thing to show depending on the person you’re showing it to, though.

For DS9, in the hands of the prophets. Doesn’t require much backstory unlike a lot of DS9, great ep, classic ethical debate/social commentary, and the majesty that is Vedek Winn.

For Voyager maybe distant origin. Great ep, doesn’t need much backstory, also great commentary on society

I really like The Drumhead from TNG. It establishes the nature of Star Trek at its most essential. It's mostly a talking episode, although there's some action with an explosion, which is perfectly average to me. It gives you a feel of the dynamics of some of the politics in the universe, which I think is a great way to get people involved. It's got one one of those great Picard speeches that puts a badmiral in their place, solving the problem non-violently. It's also a great parallel to any slippery slope security tightening after a major event happens, which is basically always a timeless message of avoiding overreaching authoritarianism at all costs.

Another TNG one I'd pick is probably The Ensigns of Command. It's another example of an episode that's mostly talking, a little bit of action, with a non-violent resolution. It's fun watching Picard come up with inane legal bullshit to deal with the very strictly by the books alien species, satisfying their requirements in a way that meets his agenda while also being within the rules.

Honestly, I could rationalize different episodes all day, but since those were the first two that came to mind, I'll just leave them at that.

I think I'd suggest a planet-of-the-week episode. From TOS, "This Side of Paradise" or "A Taste of Armageddon." From TNG, maybe "Who Watches the Watchers" or "Up the Long Ladder."

I don't watch Star Trek, but I want to pitch in that the only episode I've ever seen is Trouble With Tribbles and I enjoyed it.

In this list of suggestions, I wanted to showcase episodes that try to give a good overview of the cast and what they do and the feel of the series they are in. Note that these are what I feel are representative and relatively newbie-friendly, not necessarily the best episodes or even my favourites. So in that context, here are my picks and why:

• TOS: "The Doomsday Machine" - it showcases the Enterprise encountering an alien menace, how the crew and ship works together to defeat it, the strength of the relationship between the central trio of Kirk, Spock and McCoy and it's a great, exciting story to boot, with one of the strongest guest stars they ever had.

• TAS: "Beyond the Farthest Star" - this is one which I felt they could have done for TOS if not for the budget. It feels the closest to a live-action TOS episode.

• TNG: "The Best of Both Worlds" - especially the remastered HD movie cut. There really is no substitute for introducing people to TNG. This is the show firing on all thrusters: suspenseful, thrilling, and truly epic. It's a cracking good story, pretty self-contained, moves at a steady pace, it has great characterisation and action, and if effective, it should lead to a lot of questions about the broader universe - the Borg, Guinan, Data - you can use as a means to lead newbies to more episodes.

• DS9: "Civil Defence" - this may seem like an odd choice, being a bottle show, but DS9 was always a bit of a bottle show. I like this because it's strong on character and shows off the relationships between the characters in a crisis situation, and it brings in both Garak and Dukat, whose interactions are always a delight.

• VOY: "The Equinox" - another two-parter, but this shows off the difference between a principled and unprincipled captain and crew, a glimpse into how Janeway and Voyager could have succumbed to the dark side. (very close runner up: "Timeless", just for an example of how VOY loved to muck with time travel and alternate timelines)

• ENT: "Dead Stop" - a strange space station, Mayweather actually gets something to do (he gets killed), the need for the NX-01 to get repairs in unexplored space, and a genuine horror mystery to solve.

• DIS: "Stormy Weather" - an episode so Star Trek it couldn’t possibly be more unless it put on pointy ears and attended a convention wearing a I GROK SPOCK t-shirt. It's sciencey, it's hopeful, it's philosophical, it has kisses to continuity and even has everyone in the ensemble have a moment. And apart from that it is paced well, suspenseful and thoughtfully written. Well done all around. Honorable mention: "Magic to Make the Sanest Man Go Mad".

• LD: “wej Duj”- which I would argue is the finest Star Trek episode of the last few years and one of the best ever Star Trek episodes ever for the way it portrays the lower decks of Klingons, Vulcans, Federation and even Pakleds and… well, wait till the end credits. It also introduces fan favorite T’Lyn to the audience.

• SNW: “Memento Mori” - there’s a lot to like about the episode - the submarine warfare conceit in Star Trek space battles has always been a signature so it’s nice to see it crop up again, and you don’t even mind that the correspondence is so blatant because it’s so well executed and the keeps the tension up pretty much throughout the episode.

I’m skipping PIC because that’s not really a show for fresh eyes.

This is a great question. I like to recommend Data's Day for newbies because he sort of goes around and introduces every character, but it's a pretty unconventional episode.

I think I would go with "Doomsday Machine"!

That said, I also think Darmok would be a good fit too, the concept itself is unique in that as far as I know it's the first it was used in science fiction, but the manner in which the concept is presented "Enterprise crew encounters a problem that can't be solved with guns or technology" is pretty darn representative.

Arena.

I'm gonna have to go with "Inner Light" from TNG

It has so many typical star trek things, but it does lack a few things here and there.

It's always my go-to when someone wants an episode recommendation to see if they'll like the show.

Voyager: One Small Step

It's one of my top ten favourites, but it's also a very typical "one off" story.

"Conspiracy"

TNG, The Drumhead. The social commentary aspect of Trek without being too heavy on needing to know the SciFi lore of the series.

Just don't show them "Sub Rosa", there's no nudity at all.

I think 'The Devil in the Dark' from the Original Series would be a good option. As would "Lift Us Where Suffering Cannot Reach" from Strange New Worlds.